It is natural to ask what happens to the CY metrics for non-maximal polarized algebraic degenerations. Recall maximal degenerations require the essential skeleton to have dimension $n$. The other extreme case, where $\dim Sk(X)=0$, corresponds to degenerations with a uniform volume noncollapsing condition, and is by now quite well understood [50]: the metric limit is a Calabi-Yau variety with klt singularities. The case $0<\dim Sk(X)<n$ is expected to exhibit a mixture of both the algebraic and NA/tropical behaviours. There is no systematic theory, but the CY metrics are described in some sporadic examples [25][42]. In such examples, a key role is played by a generalized Gibbons-Hawking ansatz, which expresses CY metrics with some torus symmetry in terms of both symplectic moment coordinates and complex coordinates on the Kähler quotient. In the generic region, the asymptotic behaviour of these metrics is captured by the Calabi ansatz, which describes the metric solely in complex coordinates using the Kähler potential [42, chapter 2]. The procedure to pass from the Calabi ansatz to the generalized Gibbons-Hawking ansatz is akin to the Legendre transform.

The Calabi ansatz is as follows. Let $(Y,\Omega_{Y})$ be a compact $(n-1)$-dimensional CY manifold with an ample line bundle $E$, and let $h$ denote the Hermitian metric on $E\to Y$ whose curvature form is the Calabi-Yau metric on $Y$ in the class $c_{1}(E)$, so the length $r=h^{1/2}$ defines a function on the total space $E$. Let $\xi$ denote a local holomorphic fibre coordinate. The subset $\{0<r\lesssim 1\}\subset E$ has a nowhere vanishing holomorphic volume form $\Omega_{E}=\Omega_{Y}\wedge d\log\xi$, defined indpendent of $\xi$. Then the metric ansatz

$$\omega_{E}=dd^{c}(-\log r)^{(n+1)/n}$$

defines a CY metric on $\{0<r\lesssim 1\}\subset E$ compatible with $\Omega_{E}$.

Very little is known about the relationship between the NA MA solution and the metric degenerations, even in the aforementioned cases where the metric is well understood. Relating these is likely to require first proving some version of the comparison property, which is a major open problem. Notwithstanding all technical difficulties, we speculate the following heuristic principle: in the small $t$ limit, inside the generic region on $X_{t}$, the NA solution should approximately describe the behaviour of the Kähler potential at large scales, and obliterate the information on small compact directions. Based on this principle, we will arrive at a generalised Calabi ansatz, which is a candidate limiting description of the CY metrics in the generic region, at least in some idealized situations.

We assume the setting of section 5.3, and denote the NA MA solution as $\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$. Let $\Delta_{J}$ be an $m$-dimensional face of $Sk(X)$. We impose the simplifying assumptions:

• •

  There is no $E_{i}$ among $i\in I$ intersecting $E_{J}$ transversely in $\mathcal{X}$. Consequently, the Poincaré residue $\text{Res}_{E_{J}}(\Omega)$ is nowhere vanishing on $E_{J}$. In this situation, the complex geometric picture around $E_{J}\subset\mathcal{X}$ is modelled on the total space of the rank $m$ vector bundle $\tilde{\pi}:\oplus_{0}^{m}\mathcal{O}(E_{i})\to E_{J}$, with a trivialisation $\mathcal{O}(\sum E_{i})\simeq\mathcal{O}$ provided by the coordinate $t$. The holomorphic volume form for $|t|\ll 1$ is approximately

  $$\Omega\sim\tilde{\pi}^{*}\text{Res}_{E_{J}}(\Omega)\wedge td\log z_{0}\wedge\ldots d\log z_{m},$$

  so the holomorphic volume form $\Omega_{t}$ on $X_{t}$ is approximately

  $$\Omega_{t}\sim(-1)^{n-m}\tilde{\pi}^{*}\text{Res}_{E_{J}}\wedge d\log z_{1}\wedge\ldots d\log z_{m}.$$

• •

  For every $x\in\text{Int}(\Delta_{J})$, the class $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})\in H^{1,1}(E_{J})$ is Kähler (cf. Remark 5.1).

• •

  The NA solution $\phi_{0}$ is smooth, and in particular strictly convex on $\text{Int}(\Delta_{J})$.

We recycle the computation in section 5.2, to construct the overparametrisation $u$ of $\phi_{0}$, and produce the Hermitian metric $h_{t}$ on $(X,c_{1}(L))$, so that $h_{t}^{1/|\log|t||}$ naturally converges to $\left\lVert\cdot\right\rVert_{CY}^{2}$ in the hybrid topology. Up to $O(\frac{1}{|\log|t||})$ relative error, we have the metric asymptote (13) and the measure asymptote (14). This procedure is essentially dictated by the NA information. In particular, for each $x\in\text{Int}(\Delta_{J})$, the class

$$[-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}]=\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})\in H^{1,1}(E_{J})$$

is fixed by the NA MA solution.

Notice at this stage we still have the freedom to adjust $h_{\mathcal{L}}$ and $r_{i}$ which enter the construction in section 5.2. This information controls the small scale metric, and is not directly visible to the NA MA solution. For each $x\in\text{Int}(\Delta_{J})$, let $\phi_{x}$ be the solution to the complex MA equation on $E_{J}$:

$$\frac{\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}\right)^{n-m}}{\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}}=\frac{\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}{\int_{E_{J}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}.$$

(19)

If we wish to completely determine $\phi_{x}$ we need to fix a normalisation, but this choice is not essential. Under our hypotheses the dependence of $\phi_{x}$ on $x$ can be made smooth.

We then modify $h_{t}$ into $\tilde{h}_{t}=h_{t}e^{-2\phi_{x}}$, where $x=\text{Log}_{\mathcal{X}}(z)$ on $X_{t}$. The metric asymptote (13) then implies that on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))\subset X_{t}$ for $|t|\ll 1$, up to $O(\frac{1}{|\log|t||})$ relative error,

$$\begin{split}-dd^{c}\log\tilde{h}_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}.\end{split}$$

(20)

The normalisation ambiguity on $\phi_{x}$ is suppressed by $O(\frac{1}{|\log|t||})$ relative to the term $\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$. In particular, we see $-dd^{c}\log\tilde{h}_{t}^{1/2}$ is positive definite, so defines a Kähler metric.

We then compute its volume form up to $O(\frac{1}{|\log|t||})$ relative error:

$$\begin{split}(-dd^{c}\log\tilde{h}_{t}^{1/2})^{n}\sim&m!\det(D^{2}\phi_{0})\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}\right)^{n-m}\wedge\\ &\prod_{i=1}^{m}\frac{1}{2\pi|\log|t||}d\log|z_{i}|\wedge d\arg{z}_{i},\\ \sim m!\det(D^{2}\phi_{0})&\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\frac{\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}{\int_{E_{J}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}\wedge\\ &\prod_{i=1}^{m}\frac{\sqrt{-1}}{4\pi|\log|t||}d\log z_{i}\wedge d\log\bar{z}_{i}\\ \sim\frac{m!}{(4\pi|\log|t||)^{m}}&\det(D^{2}\phi_{0})\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\frac{\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}}{\int_{E_{J}}\sqrt{-1}^{(n-m)^{2}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}.\end{split}$$

where we use the metric asymptote (20), the construction of $\phi_{x}$ (19), and the asymptote of $\Omega_{t}$ in terms of the Poincaré residue. Now applying the PDE interpretation of NA MA equation (18), we see

$$(-dd^{c}\log\tilde{h}_{t}^{1/2})^{n}\sim\frac{(L^{n})}{(4\pi|\log|t||)^{m}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}.$$

(21)

This means the metric $-dd^{c}\log\tilde{h}_{t}^{1/2}$ is approximately Calabi-Yau up to $O(\frac{1}{|\log|t||})$ relative error in the region on $X_{t}$ corresponding to the (slightly shrinked) interior of $\Delta_{J}$. We call $-dd^{c}\log\tilde{h}_{t}^{1/2}$ the generalised Calabi ansatz, and we expect this to model the CY metric on the generic region of $X_{t}$ in the class $c_{1}(L)$ up to small error. This provides a very tight relation between the NA MA equation and the degenerating CY metrics on $X_{t}$.

We now explain how to see the Calabi ansatz as a special case. The analogue of $\mathcal{X}$ is the total space of the rank 2 vector bundle $E\oplus E^{-1}\to Y$, so the det bundle has a canonical trivialisation coordinate $t$, which defines $X_{t}\subset E\oplus E^{-1}$. Equivalently $X_{t}$ can be viewed as a submanifold of $E$. There is a holomorphic form on the total space $E\oplus E^{-1}$, given by $\Omega=\Omega_{Y}\wedge td\log z_{0}\wedge d\log z_{1}$, where $z_{1}$ is a local fibre coordinate on $E\to Y$, and $z_{0}z_{1}=t$. There is an induced holomorphic volume form $\Omega_{t}$ on $X_{t}$ defined by $\Omega=dt\wedge\Omega_{t}$, calculated to be $\Omega_{t}=(-1)^{n-1}\Omega_{Y}\wedge d\log z_{1}$, compatible up to sign with the Calabi ansatz setup. Now assume $E$ is ample, and is endowed with a Hermitian metric $h$ whose curvature form is the CY metric in $(Y,c_{1}(E))$. Denote $r=h^{1/2}$ as the length function on $E$. The analogue of $\mathcal{L}$ is $\mathcal{O}$. By the generalized Calabi ansatz prescription, we should find a function $\phi:\mathbb{R}_{+}\to\mathbb{R}$ satisfying the special case of the NA MA equation (18)

$$\phi^{\prime\prime}(\phi^{\prime})^{n-1}=\text{const},$$

and produce the metric $-\log|t|dd^{c}\phi(\frac{\log r}{\log|t|})$ on $X_{t}$. The solution $\phi(x)=x^{(n+1)/n}$ reproduces the Calabi ansatz up to scaling.